Removable singularities for subharmonic functions (Q1114045)

Let \(\Omega\) be an open set in \({\mathbb{R}}^ n\) \((n\geq 3)\) and \(S\) be a \(C^ 2\) \((n-1)\)-dimensional manifold in \(\Omega\). Let \(\alpha\in (0,n-2)\) and \(E\) be a compact subset of \(S\) of zero \(\alpha\)-dimensional Hausdorff measure. In this paper it is shown that, if \(s\) is subharmonic in \(\Omega\setminus E\) and satisfies \(s(X)\leq c[\text{dist}(X,S)]^{\alpha +2-n}\) for \(X\in \Omega \setminus S\), then \(s\) has a subharmonic extension to the whole of \(\Omega\). The sharpness of this and other related results is also established.